C H A P T E R
6Multiuser capacity and opportunistic communication
In Chapter 4, we studied several specific multiple access techniques (TDMA/FDMA, CDMA, OFDM) designed to share the channel among several users. A natural question is: what are the “optimal” multiple access schemes? To address this question, one must now step back and take a fundamental look at the multiuser channels themselves. Information theory can be generalized from the point-to-point scenario, considered in Chapter 5, to the multiuser ones, providing limits to multiuser communications and suggesting optimal multiple access strategies. New techniques and concepts such as successive cancellation, superposition coding and multiuser diversity emerge.
The first part of the chapter focuses on the uplink (many-to-one) and downlink (one-to-many) AWGN channel without fading. For the uplink, an optimal multiple access strategy is for all users to spread their signal across the entire bandwidth, much like in the CDMA system in Chapter 4. However, rather than decoding every user treating the interference from other users as noise, a successive interference cancellation (SIC) receiver is needed to achieve capacity. That is, after one user is decoded, its signal is stripped away from the aggregate received signal before the next user is decoded. A similar strategy is optimal for the downlink, with signals for the users superimposed on top of each other and SIC done at the mobiles: each user decodes the information intended for all of the weaker users and strips them off before decoding its own. It is shown that in situations where users have very disparate channels to the base-station, CDMA together with successive cancellation can offer significant gains over the conventional multiple access techniques discussed in Chapter 4.
In the second part of the chapter, we shift our focus to multiuser fading channels. One of the main insights learnt in Chapter 5 is that, for fast fading channels, the ability to track the channel at the transmitter can increase point- to-point capacity by opportunistic communication: transmitting at high rates when the channel is good, and at low rates or not at all when the channel is poor. We extend this insight to the multiuser setting, both for the uplink
229 6.1 Uplink AWGN channel
and for the downlink. The performance gain of opportunistic communication comes from exploiting the fluctuations of the fading channel. Compared to the point-to-point setting, the multiuser settings offer more opportunities to exploit. In addition to the choice of when to transmit, there is now an additional choice of which user(s) to transmit from (in the uplink) or to transmit to (in the downlink) and the amount of power to allocate between the users. This additional choice provides a further performance gain not found in the point- to-point scenario. It allows the system to benefit from a multiuser diversity effect: at any time in a large network, with high probability there is a user whose channel is near its peak. By allowing such a user to transmit at that time, the overall multiuser capacity can be achieved.
In the last part of the chapter, we will study the system issues arising from the implementation of opportunistic communication in a cellular system. We use as a case study IS-856, the third-generation standard for wireless data already introduced in Chapter 5. We show how multiple antennas can be used to further boost the performance gain that can be extracted from opportunistic communication, a technique known as opportunistic beamforming. We distill the insights into a new design principle for wireless systems based on opportunistic communication and multiuser diversity.
6.1 Uplink AWGN channel
6.1.1 Capacity via successive interference cancellation
The baseband discrete-time model for the uplink AWGN channel with two users (Figure 6.1) is
Figure 6.1 Two-user uplink.
y m = x1 m + x2 m + w m |
(6.1) |
where w m 0 N0 is i.i.d. complex Gaussian noise. User k has an average power constraint of Pk joules/symbol (with k = 1 2).
In the point-to-point case, the capacity of a channel provides the performance limit: reliable communication can be attained at any rate R < C; reliable communication is impossible at rates R > C. In the multiuser case, we should extend this concept to a capacity region : this is the set of all pairs R1 R2 such that simultaneously user 1 and 2 can reliably communicate at rate R1 and R2, respectively. Since the two users share the same bandwidth, there is naturally a tradeoff between the reliable communication rates of the users: if one wants to communicate at a higher rate, the other user may need to lower its rate. For example, in orthogonal multiple access schemes, such as OFDM, this tradeoff can be achieved by varying the number of sub-carriers allocated to each user. The capacity region characterizes the optimal tradeoff achievable by any multiple access scheme. From this
230 |
Multiuser capacity and opportunistic communication |
capacity region, one can derive other scalar performance measures of interest. For example:
• The symmetric capacity:
is the maximum common rate at which both the users can simultaneously reliably communicate.
• The sum capacity:
C |
sum |
= |
max |
|
R1 + R2 |
(6.3) |
|
R1 R2 |
|
|
|
|
|
|
|
|
|
|
is the maximum total throughput that can be achieved.
Just like the capacity of the AWGN channel, there is a very simple characterization of the capacity region of the uplink AWGN channel: this is the set of all rates R1 R2 satisfying the three constraints (Appendix B.9 provides a formal justification):
|
|
|
|
|
|
|
|
|
|
|
|
|
+ P2 |
|
|
|
R1 |
< log |
1 |
|
P1 |
|
(6.4) |
|
|
N0 |
|
R2 |
< log |
1 + |
|
|
|
(6.5) |
|
N0 |
|
|
|
|
P1+P2 |
|
(6.6) |
|
R1 + R2 < log 1 + |
|
|
N0 |
|
The capacity region is the pentagon shown in Figure 6.2. All the three constraints are natural. The first two say that the rate of the individual user cannot exceed the capacity of the point-to-point link with the other user absent from
Figure 6.2 Capacity region of the two-user uplink AWGN channel.
|
|
|
R2 |
log 1 + |
P2 |
B |
|
|
|
N0 |
|
P2 |
C |
log 1 +P1 + N |
0 |
A |
R1
log 1 + P |
|
P1 |
P1 |
|
|
|
|
log 1 + |
|
|
|
|
2 |
+ N |
N |
0 |
|
|
0 |
|
|
231 |
6.1 Uplink AWGN channel |
the system (these are called single-user bounds). The third says that the total throughput cannot exceed the capacity of a point-to-point AWGN channel with the sum of the received powers of the two users. This is indeed a valid constraint since the signals the two users send are independent and hence the power of the aggregate received signal is the sum of the powers of the individual received signals.1 Note that without the third constraint, the capacity region would have been a rectangle, and each user could simultaneously transmit at the point-to-point capacity as if the other user did not exist. This is clearly too good to be true and indeed the third constraint says this is not possible: there must be a tradeoff between the performance of the two users.
Nevertheless, something surprising does happen: user 1 can achieve its single-user bound while at the same time user 2 can get a non-zero rate; in fact as high as its rate at point A, i.e.,
2 = |
|
|
+ |
N0 |
− |
|
+ N0 |
= |
|
+ P1 |
+ N0 |
|
R |
log |
1 |
|
P1 + P2 |
|
log 1 |
|
P1 |
|
log 1 |
|
|
P2 |
(6.7) |
|
|
|
|
|
|
|
|
|
How can this be achieved? Each user encodes its data using a capacityachieving AWGN channel code. The receiver decodes the information of both the users in two stages. In the first stage, it decodes the data of user 2, treating the signal from user 1 as Gaussian interference. The maximum rate user 2 can achieve is precisely given by (6.7). Once the receiver decodes the data of user 2, it can reconstruct user 2’s signal and subtract it from the aggregate received signal. The receiver can then decode the data of user 1. Since there is now only the background Gaussian noise left in the system, the maximum rate user 1 can transmit at is its single-user bound log 1 + P1/N0 . This receiver is called a successive interference cancellation (SIC) receiver or simply a successive cancellation decoder. If one reverses the order of cancellation, then one can achieve point B, the other corner point. All the other rate points on the segment AB can be obtained by time-sharing between the multiple access strategies in point A and point B. (We see in Exercise 6.7 another technique called rate-splitting that also achieves these intermediate points.)
The segment AB contains all the “optimal” operating points of the channel, in the sense that any other point in the capacity region is component-wise dominated by some point on AB. Thus one can always increase both users’ rates by moving to a point on AB, and there is no reason not to.2 No such domination exists among the points on AB, and the preferred operating point depends on the system objective. If the goal of the system is to maximize the sum rate, then any point on AB is equally fine. On the other hand, some operating points are not fair, especially if the received power of one user is
1 This is the same argument we used for deriving the capacity of the MISO channel in Section 5.3.2.
2 In economics terms, the points on AB are called Pareto optimal.
232 Multiuser capacity and opportunistic communication
much larger than the other. In this case, consider operating at the corner point in which the strong user is decoded first: now the weak user gets the best possible rate.3 In the case when the weak user is the one further away from the base-station, it is shown in Exercise 6.10 that this decoding order has the property of minimizing the total transmit power to meet given target rates for the two users. Not only does this lead to savings in the battery power of the users, it also translates to an increase in the system capacity of an interference-limited cellular system (Exercise 6.11).
6.1.2 Comparison with conventional CDMA
There is a certain similarity between the multiple access technique that achieves points A and B, and the CDMA technique discussed in Chapter 4. The only difference is that in the CDMA system described there, every user is decoded treating the other users as interference. This is sometimes called a conventional or a single-user CDMA receiver. In contrast, the SIC receiver is a multiuser receiver: one of the users, say user 1, is decoded treating user 2 as interference, but user 2 is decoded with the benefit of the signal of user 1 already removed. Thus, we can immediately conclude that the performance of the conventional CDMA receiver is suboptimal; in Figure 6.2, it achieves the point C which is strictly in the interior of the capacity region.
The benefit of SIC over the conventional CDMA receiver is particularly significant when the received power of one user is much larger than that of the other: by decoding and subtracting the signal of the strong user first, the weaker user can get a much higher data rate than when it has to contend with the interference of the strong user (Figure 6.3). In the context of a cellular system, this means that rather than having to keep the received powers of all users equal by transmit power control, users closer to the base-station can be allowed to take advantage of the stronger channel and transmit at a higher rate while not degrading the performance of the users in the edge of the cell. With a conventional receiver, this is not possible due to the near–far problem. With the SIC, we are turning the near–far problem into a near–far advantage. This advantage is less apparent in providing voice service where the required data rate of a user is constant over time, but it can be important for providing data services where users can take advantage of the higher data rates when they are closer to the base-station.
6.1.3 Comparison with orthogonal multiple access
How about orthogonal multiple access techniques? Can they be information theoretically optimal? Consider an orthogonal scheme that allocates a fraction
3 This operating point is said to be max–min fair.
234
Figure 6.4 Performance of orthogonal multiple access compared to capacity. The SNRs of the two users are:
P1/N0 = 0 dB and
P2/N0 = 20 dB. Orthogonal multiple access achieves the sum capacity at exactly one point, but at that point the weak user 1 has hardly any rate and it is therefore a highly unfair operating point. Point A gives the highest possible rate to user 1 and is most fair.
Multiuser capacity and opportunistic communication
R2 ( bits / s / Hz )
BSum capacity
6.66achieved here
R1 ( bits / s / Hz ) 0.014 0.065 1
can approach this performance for the weak user only by nearly sacrificing all the rate of the strong user. Here again, as in the comparison with CDMA, SIC’s advantage is in exploiting the proximity of a user to the base-station to give it high rate while protecting the far-away user.
6.1.4 General K -user uplink capacity
We have so far focused on the two-user case for simplicity, but the results extend readily to an arbitrary number of users. The K-user capacity region is described by 2K − 1 constraints, one for each possible non-empty subset of users:
|
k |
Pk |
|
|
k Rk < log 1 + |
|
|
N0 |
|
for all 1 K |
(6.10) |
The right hand side corresponds to the maximum sum rate that can be achieved by a single transmitter with the total power of the users in and with no other users in the system. The sum capacity is
Csum = log 1 + |
K |
Pk |
|
k=1 |
|
bits/s/Hz |
|
N0 |
|
(6.11) |
It can be shown that there are exactly K! corner points, each one corresponding to a successive cancellation order among the users (Exercise 6.9).
The equal received power case (P1 = = PK = P) is particularly simple. The sum capacity is
Csum = log 1 + |
KP |
|
(6.12) |
N0 |
235 6.2 Downlink AWGN channel
The symmetric capacity is |
|
|
|
|
|
Csym = |
1 |
· log 1 + |
KP |
|
(6.13) |
|
|
K |
N0 |
This is the maximum rate for each user that can be obtained if every user operates at the same rate. Moreover, this rate can be obtained via orthogonal multiplexing: each user is allocated a fraction 1/K of the total degrees of freedom.4 In particular, we can immediately conclude that under equal received powers, the OFDM scheme considered in Chapter 4 has a better performance than the CDMA scheme (which uses conventional receivers.)
Observe that the sum capacity (6.12) is unbounded as the number of users grows. In contrast, if the conventional CDMA receiver (decoding every user treating all other users as noise) is used, each user will face an interference from K − 1 users of total power K − 1 P, and thus the sum rate is only
|
K · log 1 + |
P |
bits/s/Hz |
(6.14) |
|
|
|
K − 1 P + N0 |
which approaches |
|
|
|
K · |
P |
log2 e ≈ log2 e = 1 442 bits/s/Hz |
(6.15) |
|
|
K − 1 P + N0 |
as K → . Thus, the total spectral efficiency is bounded in this case: the growing interference is eventually the limiting factor. Such a rate is said to be interference-limited.
The above comparison pertains effectively to a single-cell scenario, since the only external effect modeled is white Gaussian noise. In a cellular network, the out-of-cell interference must be considered, and as long as the out-of-cell signals cannot be decoded, the system would still be interference-limited, no matter what the receiver is.
6.2 Downlink AWGN channel
The downlink communication features a single transmitter (the base-station) sending separate information to multiple users (Figure 6.5). The baseband downlink AWGN channel with two users is
yk m = hkx m + wk m k = 1 2 |
(6.16) |
where wk m 0 N0 is i.i.d. complex Gaussian noise and yk m is the received signal at user k at time m, for both the users k = 1 2. Here hk is
4 This fact is specific to the AWGN channel and does not hold in general. See Section 6.3.
236 |
Multiuser capacity and opportunistic communication |
the fixed (complex) channel gain corresponding to user k. We assume that hk is known to both the transmitter and the user k (for k = 1 2). The transmit signal x m has an average power constraint of P joules/symbol. Observe the difference from the uplink of this overall constraint: there the power restrictions are separate for the signals of each user. The users separately decode their data using the signals they receive.
As in the uplink, we can ask for the capacity region , the region of the ratesR1 R2 , at which the two users can simultaneously reliably communicate. We have the single-user bounds, as in (6.4) and (6.5),
Rk < log 1 + |
P hk |
|
2 |
|
|
|
|
|
k = 1 2 |
(6.17) |
N0 |
|
|
Figure 6.5 Two-user downlink.
This upper bound on Rk can be attained by using all the power and degrees of freedom to communicate to user k (with the other user getting zero rate). Thus, we have the two extreme points (with rate of one user being zero) in Figure 6.6. Further, we can share the degrees of freedom (time and bandwidth) between the users in an orthogonal manner to obtain any rate pair on the line joining these two extreme points. Can we achieve a rate pair outside this triangle by a more sophisticated communication strategy?
6.2.1 Symmetric case: two capacity-achieving schemes
To get more insight, let us first consider the symmetric case where h1 = h2 . In this symmetric situation, the SNR of both the users is the same. This means that if user 1 can successfully decode its data, then user 2 should also be
R2
log 1+ h22 P N0
Figure 6.6 The capacity region of the downlink with two users having symmetric AWGN channels, i.e., h1 = h2 .
R1
h2 P log 1+ 1
N0
237 |
6.2 Downlink AWGN channel |
able to decode successfully the data of user 1 (and vice versa). Thus the sum information rate must also be bounded by the single-user capacity:
R1 + R2 < log 1 + P h1 2 (6.18)
N0
Comparing this with the single-user bounds in (6.17) and recalling the symmetry assumption h1 = h2 , we have shown the triangle in Figure 6.6 to be the capacity region of the symmetric downlink AWGN channel.
Let us continue our thought process within the realm of the symmetry assumption. The rate pairs in the capacity region can be achieved by strategies used on point-to-point AWGN channels and sharing the degrees of freedom (time and bandwidth) between the two users. However, the symmetry between the two channels (cf. (6.16)) suggests a natural, and alternative, approach. The main idea is that if user 1 can successfully decode its data from y1, then user 2, which has the same SNR, should also be able to decode the data of user 1 from y2. Then user 2 can subtract the codeword of user 1 from its received signal y2 to better decode its own data, i.e., it can perform successive interference cancellation. Consider the following strategy that superposes the signals of the two users, much like in a spread-spectrum CDMA system. The transmit signal is the sum of two signals,
where xk m is the signal intended for user k. The transmitter encodes the information for each user using an i.i.d. Gaussian code spread on the entire bandwidth (and powers P1 P2, respectively, with P1 + P2 = P). User 1 treats the signal for user 2 as noise and can hence be communicated to reliably at a rate of
|
|
1 |
|
|
|
P |
h |
1 |
2 |
|
|
|
|
|
|
|
|
|
|
|
R1 = log |
+ |
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
P2 |
|
h1 2 |
+ |
N0 |
|
|
|
|
|
|
|
|
|
= |
|
|
+ |
|
|
|
|
|
2 |
|
− |
|
|
+ |
N0 |
2 |
|
|
|
|
|
|
|
N0 |
|
|
|
|
|
log |
1 |
|
P1 + P2 h1 |
|
|
log |
1 |
|
P2 h1 |
|
(6.20) |
|
|
|
|
|
|
|
|
|
User 2 performs successive interference cancellation: it first decodes the data of user 1 by treating x2 as noise, subtracts the exactly determined (with high probability) user 1 signal from y2 and extracts its data. Thus user 2 can support reliably a rate
|
2 = |
|
|
+ |
N0 |
2 |
|
|
R |
|
log |
1 |
|
P2 h2 |
|
(6.21) |
|
|
|
|
This superposition strategy is schematically represented in Figures 6.7 and 6.8. Using the power constraint P1 + P2 = P we see directly from (6.20) and (6.21) that the rate pairs in the capacity region (Figure 6.6) can be achieved by this strategy as well. We have hence seen two coding schemes for the
